Range of operator is dense in $L_p$

Question

$M_au=a\cdot u, \ u\in dom(M_a)$

$dom(M_a)=\{u\in L_p(\Omega): au\in L_p(\Omega)\}.$

I have to prove that $ran(M_a)$ is dense in $L_p$ provided that $a(x) \neq 0$ a.e. I have never shown that something is dense in something, so I don't know how to start. I only know that I have to find sequence from $ran(M_a)$ which is convergent to some element from $L_p$.

Thanks in advance for any tips

Answer 1

Hint: for any $f \in L_p$ and $\varepsilon > 0$, function $$g_\varepsilon(x)= \begin{cases} f(x),\ |a(x)| \geqslant \varepsilon\\ 0,\ |a(x)| < \varepsilon \end{cases} $$ is in $\operatorname{ran(M_a)}$.